6.10 Coupled Cluster Active Space Methods 6.10.3 VQCCD 6.10.5 Local Pair Models for Valence Correlations Beyond Doubles

6.10.4 CCVB-SD

Working with Prof. Head-Gordon at Berkeley, Dr. D. W. Small and Joonho Lee have developed and implemented a novel single-reference coupled-cluster method with singles and doubles, called CCVB-SD.^{893, 418} CCVB-SD improves upon a more crude model CCVB (Section 6.16.2) and can be considered a simple modification to restricted CCSD (RCCSD). CCVB-SD inherits good properties from CCVB and RCCSD; it is spin-pure, size-extensive, and capable of breaking multiple bonds as long as only the valence space is correlated. It is a full doubles model and thus scales $\mathcal{O}(N^{6})$ . However, its energy is invariant under rotations in occupied space and virtual space, which makes it much more black-box than CCVB. Its energy function follows

E_{\mathrm{CCVB-SD}}=\left\langle{\Phi_{0}\left({1+\hat{\Lambda}}\right)\left|% {\hat{H}}\right|\left(\exp\left({\hat{T}}\right)-\hat{\text{I}}_{\text{S}}% \frac{\hat{Q}^{2}}{2}\right)\Phi_{0}}\right\rangle_{C}

(6.42)

where $\hat{\text{I}}_{\text{S}}$ is a singlet projection operator and $\hat{Q}$ is a quintet doubles operator. Unlike QCCD, CCVB-SD improves the right eigenfunction while leaving the left eigenfunction unchanged. The quintet term in Eq. (6.42) represents approximate connected quadruples which are responsible for describing strong correlation. The cost of CCVB-SD is only twice as expensive as RCCSD, and it is better suited for strong correlation than QCCD/VQCCD in the sense that the method becomes exact at the dissociation limits of most multiple bond breaking whereas QCCD does not except special cases.

Although CCVB-SD can be used without the active space constraints, we recommend that users use it with the valence active space in general. For benchmarking purposes, using a minimal basis will automatically provide the valence space correctly with frozen cores. Both the energy and nuclear gradients of CCVB-SD are available through CCMAN2.

It should be noted that there is no orbital optimization implemented for CCVB-SD at the moment. This means that using basis sets larger than minimal basis requires choosing right valence orbitals to use. Therefore, we recommend that users run GVB-PP (or CCVB) to obtain orbitals to begin with. Orbital optimization (i.e. CCVB-OD) will soon be implemented and running CCVB-OD will be much more black-box than CCVB-SD as it does not require selecting proper valence space orbitals.

Furthermore, CCVB-SD can be applied to only closed-shell molecules at the moment. The extension to open-shell molecules is under development.

Example 6.27 A CCVB-SD force calculation of benzene in a minimal basis.

$comment
CCVB-SD job for benzene computing energy+gradients.
It will also print out natural orbital occupation numbers (NOONs)
$end

$molecule
   0 1
   C    0.000000    0.698200    0.000000
   C    0.000000   -0.698200    0.000000
   C    1.209318    1.396400    0.000000
   C    1.209318   -1.396400    0.000000
   C    2.418636    0.698200    0.000000
   C    2.418636   -0.698200    0.000000
   H   -0.931410    1.235950    0.000000
   H   -0.931410   -1.235950    0.000000
   H    1.209318    2.471900    0.000000
   H    1.209318   -2.471900    0.000000
   H    3.350046    1.235950    0.000000
   H    3.350046   -1.235950    0.000000
$end

$rem
   JOBTYPE         = force
   BASIS           = sto-3g
   METHOD          = ccvbsd
   THRESH          = 14
   SCF_ALGORITHM   = gdm
   SCF_CONVERGENCE = 10
   CC_REF_PROP     = true
   SYMMETRY        = false
   SYM_IGNORE      = true
$end

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6.10.4 CCVB-SD